Find third vertex of triangle on the Cartesian plane

Question

I have a triangle on the cartesian plane where I know the following: $$ A = (x_1,y_1), B = (x_2,y_2), C = (x_3,y_3) $$

I want to find the possible locations for $C$.

I know the location of $A$ and $B$.

I know that I want the angle $ABC$ will be $90^{\mkern1mu\mathrm{o}}$.

and for narrowing the search down, I know that if: $$ AB = x $$ then:

$$ AC = 2x $$ example triangle

Answer 1

There can be no solution: if $\widehat{ACB}$ is a right angle, C is on the circle with diameter $AB$, and if $AC=2AB$, it is on the circle with centre $A$ and radius $2AB$. These two circles have no intersection.